Optimal. Leaf size=113 \[ -\frac {1}{9} \left (\frac {1-x^3}{x^3+1}\right )^{3/2} \left (x^3+1\right )^3-\frac {1}{6} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )^2+\frac {1}{2} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )-\frac {1}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{x^3+1}}\right ) \]
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Rubi [A] time = 0.06, antiderivative size = 113, normalized size of antiderivative = 1.00, number of steps used = 5, number of rules used = 5, integrand size = 23, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.217, Rules used = {1960, 463, 455, 385, 204} \[ -\frac {1}{9} \left (\frac {1-x^3}{x^3+1}\right )^{3/2} \left (x^3+1\right )^3-\frac {1}{6} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )^2+\frac {1}{2} \sqrt {\frac {1-x^3}{x^3+1}} \left (x^3+1\right )-\frac {1}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{x^3+1}}\right ) \]
Antiderivative was successfully verified.
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Rule 204
Rule 385
Rule 455
Rule 463
Rule 1960
Rubi steps
\begin {align*} \int x^8 \sqrt {\frac {1-x^3}{1+x^3}} \, dx &=-\left (\frac {4}{3} \operatorname {Subst}\left (\int \frac {x^2 \left (-1+x^2\right )^2}{\left (-1-x^2\right )^4} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\right )\\ &=-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3-\frac {2}{9} \operatorname {Subst}\left (\int \frac {x^2 \left (6-6 x^2\right )}{\left (-1-x^2\right )^3} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=-\frac {1}{6} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )^2-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3+\frac {1}{18} \operatorname {Subst}\left (\int \frac {12-24 x^2}{\left (-1-x^2\right )^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=\frac {1}{2} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {1}{6} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )^2-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3+\frac {1}{3} \operatorname {Subst}\left (\int \frac {1}{-1-x^2} \, dx,x,\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ &=\frac {1}{2} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )-\frac {1}{6} \sqrt {\frac {1-x^3}{1+x^3}} \left (1+x^3\right )^2-\frac {1}{9} \left (\frac {1-x^3}{1+x^3}\right )^{3/2} \left (1+x^3\right )^3-\frac {1}{3} \tan ^{-1}\left (\sqrt {\frac {1-x^3}{1+x^3}}\right )\\ \end {align*}
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Mathematica [A] time = 0.04, size = 98, normalized size = 0.87 \[ \frac {\sqrt {\frac {1-x^3}{x^3+1}} \sqrt {x^3+1} \left (6 \sqrt {1-x^3} \sin ^{-1}\left (\frac {\sqrt {1-x^3}}{\sqrt {2}}\right )+\sqrt {x^3+1} \left (2 x^9-5 x^6+7 x^3-4\right )\right )}{18 \left (x^3-1\right )} \]
Antiderivative was successfully verified.
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fricas [A] time = 0.44, size = 65, normalized size = 0.58 \[ \frac {1}{18} \, {\left (2 \, x^{9} - x^{6} + x^{3} + 4\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - \frac {1}{3} \, \arctan \left (\frac {{\left (x^{3} + 1\right )} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}} - 1}{x^{3}}\right ) \]
Verification of antiderivative is not currently implemented for this CAS.
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giac [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{8} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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maple [A] time = 0.07, size = 80, normalized size = 0.71 \[ -\frac {\sqrt {-\frac {x^{3}-1}{x^{3}+1}}\, \sqrt {-\left (x^{3}+1\right ) \left (x^{3}-1\right )}\, \arcsin \left (x^{3}\right )}{6 \left (x^{3}-1\right )}+\frac {\left (2 x^{6}-3 x^{3}+4\right ) \left (x^{3}+1\right ) \sqrt {-\frac {x^{3}-1}{x^{3}+1}}}{18} \]
Verification of antiderivative is not currently implemented for this CAS.
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maxima [F] time = 0.00, size = 0, normalized size = 0.00 \[ \int x^{8} \sqrt {-\frac {x^{3} - 1}{x^{3} + 1}}\,{d x} \]
Verification of antiderivative is not currently implemented for this CAS.
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mupad [B] time = 2.66, size = 101, normalized size = 0.89 \[ \frac {2\,\sqrt {-\frac {x^3-1}{x^3+1}}}{9}-\frac {\mathrm {atan}\left (\sqrt {-\frac {x^3-1}{x^3+1}}\right )}{3}+\frac {x^3\,\sqrt {-\frac {x^3-1}{x^3+1}}}{18}-\frac {x^6\,\sqrt {-\frac {x^3-1}{x^3+1}}}{18}+\frac {x^9\,\sqrt {-\frac {x^3-1}{x^3+1}}}{9} \]
Verification of antiderivative is not currently implemented for this CAS.
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sympy [F(-1)] time = 0.00, size = 0, normalized size = 0.00 \[ \text {Timed out} \]
Verification of antiderivative is not currently implemented for this CAS.
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